From an interview I read, but can't remember the source:

Alex Kurtzman: [Paul Giamatti] was doing press for The Holdovers, and when he was asked about the part that he would most want to play, he said, “I want to play a Klingon on Star Trek.”… So we called his agent and said, “Was that a thing he just said, was that a bit?”… And she called back and she said, “He’d love to sit down with you. We got on a Zoom with him and he cried.

[...]

Noga Landau: He also chose which role he wanted to play. We came to him and we said, “You could be this guy or that guy, or that guy.” And he thought about it and he came back and told us that the part he plays is the one he wanted.

Alex Kurtzman: We thought it was going to be like one episode because his schedule is so busy, and he was like, “No, I want to play the villain.” We’re like, “That’s the whole season.” He was like, “Great, let’s do it!”

Like... no one's stopping you from being a Klingon on Star Trek, Paul Giamatti.

The villain, no less!